Lazy Automata Techniques for WS1S

Abstract

Abstract. We present a new decision procedure for the logic WS1S. It originates from the classical approach, which first builds an automaton accepting all models of a formula and then tests whether its language is empty. The main novelty is to test the emptiness on the fly, while constructing a symbolic, term-based representation of the automaton, and prune the constructed state space from parts irrelevant to the test. The pruning is done by a generalization of two techniques used in antichain-based language inclusion and universality checking of finite automata: subsumption and early termination. The richer structure of the WS1S decision problem allows us, however, to elaborate on these techniques in novel ways. Our experiments show that the proposed approach can in many cases significantly outperform the classical decision procedure (implemented in the MONA tool) as well as recently proposed alternatives.

Automata Terms in a Lazy WSkS Decision Procedure

“…The next technique we use is based on pruning out parts of a search space that are subsumed by other parts. In particular, we generalize (in a similar way as we did for WS1S in our previous work [29]) the concept used in antichain algorithms for efficiently deciding language inclusion and universality of finite word and tree automata [22,23,24,25]. Although the problems are in general computationally infeasible (they are PSPACEcomplete for finite word automata and EXPTIME-complete for finite tree automata), antichain algorithms can solve them efficiently in many practical cases.…”

“…On them, our tool is quite slower than M , which is not much surprising given the amount of optimisations built into M (for instance, for the benchmarks from [5], M on average took 0.1 s, while we timeouted). Next, we identified several parametric families of formulae (adapted from [29]), such as, e.g., ϕ horn n ≡ ∃X. ∀X 1 .…”

“…This paper is a continuation of our work on WS1S, which started by [42], where we discussed a basic idea of generalizing the antichain techniques to a WS1S decision procedure. In [29], we then presented a complete WS1S decision procedure based on these ideas that is capable to rival M on already interesting benchmarks. The work in [43] presents a decision procedure that, although phrased differently, is in essence fairly similar to that of [29].…”

“…In [29], we then presented a complete WS1S decision procedure based on these ideas that is capable to rival M on already interesting benchmarks. The work in [43] presents a decision procedure that, although phrased differently, is in essence fairly similar to that of [29]. This paper generalizes [29] to WS2S.…”

“…We therefore cannot define the semantics of a state term as a tree language, and so we cannot define the semantics of an automata term as the union of the languages of its state sub-terms. This problem seems critical at first because without a sensible notion of the meaning of terms, a straightforward generalization of the algorithm of [29] to trees does not seem possible. The solution we present here is based on defining the semantics of terms via the automata constructions they represent rather then as functions of languages of their sub-terms.…”